テイラーの冪級数

1
ͳͥϕΩ‫͔਺ڃ‬ʁ
f (z) ͷۙࣅ
f (z) =
৒༨
”Ԡ༻਺ֶ̏ʵෳૉؔ਺࿦”
”ࢿྉ̒”
3.1
3.2
3.3
3.4
਺ྻɾ‫਺ڃ‬ɾऩଋ൑ఆ (91–99)ɺ
ϕΩ‫( ਺ڃ‬100–105)ɺ
ςΠϥʔ‫ͱ਺ڃ‬ϚΫϩʔϦϯ‫( ਺ڃ‬pp.112–119)
an z n = n→∞
lim
n=0
∞
ak z k
k=n
n
ak z k
k=0
n
= f (z) −
෦෼࿨ͷ‫ݶۃ‬
ak z k → 0
k=0
෦෼༗‫ݶ‬࿨ʹΑΓʮ๬Ή͚ͩͷਫ਼౓ʯͷۙࣅ͕ཧ࿦্Մೳ
ϕΩ‫਺ڃ‬͸߲ผඍ෼Մೳɺ߲ผੵ෼Մೳ
f (z) =
n
an z n
f (z) =
n
nan z n−1
ऩଋԁ಺෦͸୯࿈݁ྖҬɻͦͷதͰ
z0 ͱ z1 Λ݁Ϳ೚ҙͷ‫ܦ‬࿏ C ʹର͠
C
ϕΩ‫Ͱ਺ڃ‬༩͑ΒΕΔؔ਺ (106–110)
∞
f (z)dz =
z1
z0
f (z)dz =
n
z1
z0
an z n dz
ʁ ղੳؔ਺͸ͦ͏ͨ͠ʮϕΩ‫਺ڃ‬ʯͰද‫͖Ͱݱ‬Δ͔ʁ
˰ ςΠϥʔͷఆཧʢyes, where ϕΩ‫਺ڃ‬ͷ ऩଋԁ಺෦ ʹ͓͍ͯʣ
ʢ։ԁ൘ʣ
͞Βʹɼ
a0
f (z)
a1
=
+
+ a2 + a3 (z − a) + ...
2
2
(z − a)
(z − a)
z−a
ϩʔϥϯ‫਺ཹͱ਺ڃ‬ͷ࿩΁ɽ
ɽ
ɽ
1
2
2
‫ͱ਺ڃ‬ϕΩ‫ؔ͢ʹ਺ڃ‬Δඞཁࣄ߲
∞
ෳૉ਺ͷ‫਺ڃ‬ɿ
෦෼࿨ɿ
n
‫ج‬ຊྻ {zn }n ɿ
∀ > 0 ∃ N s.t. p, q > N ˰ |zp − zq | < (zn = xn + iyn )
n=0
n
zk =
k=0
xk + i
k=0
n
yk
࣮෦ͱ‫ڏ‬෦ɼ֤ʑͷ෦෼࿨
i2
4
n
i
఺ྻ sn =
2
+
i3
8
ऩଋ఺ྻ ˱ ‫ج‬ຊྻ ʢ࣮਺ʹର͢Δٞ࿦ͷ֦ுʣ
k=0
i, −1, −i, 1, i... ͷมಈΛ‫ؚ‬Έ
1 − in+1
:
1−i
༗ք͕ͩऩଋ͠ͳ͍ʢൃࢄʣ
n+1
1 − 2i n+1
1
+ ... = n→∞
lim
=
1 − 2i
1 − 2i
ྫɿ 1 + i + i2 + i3 + ... = n→∞
lim
1 + 2i +
Cauchy ͷऩଋ൑ఆ
2.1
α ʹऩଋ͢Δͱ͢Δɽ
ཁ‫ٻ‬ਫ਼౓ /2 ʹର͠ n > N ˰ |zn − α| < /2
for p, q > N , |zp − zq | ≤ |zp − α| + |zq − α| < ‫{ ʹٯ‬zn = an + ibn }n ͕‫ج‬ຊྻͱ͢Δɽॴ༩ͷ > 0
:
ʹର͠ɼn, m > N ˰ |zn − zm | < ͳΔ N ͕ଘࡏɽ
|an − am |, |bn − bm | ≤ |zn − zm | ͔ͩΒ
1
i 1
i
1
i
,
, − , ...
1, , − , − ,
2
4
8 16 32
64
lim sn = 0
{an }n , {bn }n ͸࣮਺ͷ‫ج‬ຊྻͰऩଋ͢Δ
n→∞
n ͕େʢn > N ʣʹͳΔͱɼ‫఺ݪ‬ͷճΓͷ‫͍ڱ‬ൣғʢখ
ԁ൫ʣ಺෦ʹ sn , sm (m > n > N ) ͸ີूͯ͠ग़‫ݱ‬ɽ
ಛʹɼsn , sm ؒͷ‫཭ڑ‬΋খɽ
ie. ‫ج‬ຊྻɿ |sn − sm | < whenever m > n > N
∞
‫਺ڃ‬
n=0
n
෦෼࿨
‫͕਺ڃ‬ऩଋɿ ෦෼࿨͕ෳૉ਺ͷ఺ྻͱͯ͠‫ݶۃ‬஋Λ࣋ͭ
⎞
⎛
lim
n→∞
n
k=0
zk = n→∞
lim ⎝
n
xk + i
k=0
n
k=0
(∗)
yk ⎠ = n→∞
lim
n
k=0
Αͬͯɼ
z xk + in→∞
lim
z ʹऩଋ
෦෼࿨ͱ z ͷࠩʢ৒༨ʣ͕‫ݶ‬Γͳ͘খʹ
= 0 ˱ n→∞
lim
˱
zk ͕‫ج‬ຊྻ
∀ > 0, ∃ N s.t. N < p < q ˰
yk
n=0
−
zn ͕ऩଋ
k=0
n
lim zk
n→∞ k=0
zn = z ˱
k=0
n
͜͜Ͱɼ఺ྻͷऩଋ n→∞
lim sn = s ˱ n→∞
lim |sn − s| = 0
∞
Αͬͯɼlim
z = lim
a + i lim
b
n n
n n
n n
∞
zk → 0
k=n
{zn = an + ibn }n ͕‫ج‬ຊྻ ͱ͢Δ
Fn = {zk }k≥n ɿ༗ք (*)ɽ࣮෦ͱ‫ڏ‬෦ͷ۠ؒ In , Jn
In = [ inf ak , sup ak ], Jn = [ inf bk , sup bk ]ɿ ༗ք
k≥n
k≥n
k≥n
k≥n
۠ؒ௕ |In | = sup |ap − aq |ɿn Ҏ߱ͷ ap , aq ͷࠩͷ্‫ݶ‬ɽ
p,q≥n
‫਺ڃ‬ʢ఺ྻʣ͕ൃࢄ ⇔ ‫਺ڃ‬ʢ఺ྻʣ͕ऩଋ͠ͳ͍
def
n Λे෼େʹͱΕ͹͍͘ΒͰ΋୹͘Ͱ͖Δɽ lim |In | = 0ɽ
∞
n→∞
{In }n ͸୯ௐ‫ݮ‬গྻɽ Αͬͯ, lim In = ∩ In = {a}
n→∞
n=0
‫ڏ‬෦ͷ۠ؒ Jn ΋ಉ༷ʹ b ΛఆΊɼ݁‫ہ‬ɼ lim zn = a + bi
(*) zn = an + ibn → w = a + ib ˰ an → a, bn → b
n→∞
|a − an |, |b − bn | < |zn − w| = |(an − a) + i(bn − b)| → 0
ͭ·Γɼ|xn − a| → 0, |yn − b| → 0
3
4
q
k=p+1
zk <
ྫɿ ‫ز‬Կ‫਺ڃ‬
2.2
∞
q =
n
n=0
∞
n niθ
r e
, q = reiθ .
n=0
q = 1 + q + ···q , q
k
n
k=0
So, (1 − q)
qk =
k=0
1 − rn+1 e(n+1)θi
1−q
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
n
2
q = q + q + ···q
k
k=0
n
k
n+1
q = 1 − q n+1
1
r < 1 Ͱr
→ 0ɽ
ʹऩଋ
1−q
1 − e(n+1)iθ
, θ = 0
r = 1 Ͱൃࢄɿ as
1−q
r > 1 Ͱɼൃࢄ
্‫͔ه‬Β΋Θ͔Δ͕ɼ ऩଋ‫਺ڃ‬
zn Ͱ͸ɼ֤߲ zn → 0
∞
f (z) =
n
‫ز‬Կ‫| Ͱ਺ڃ‬q| = 1 ͷͱ͖ɿ ֤߲ |q n | = 1 →
\ 0 Ͱൃࢄ
|zn | ≤ bn ʢӈล͸࣮਺ʣɼ
n
n
m
bn < ∞ ˰ k=n
zn ͸ऩଋ ʢCauchyʣɽ ઈରऩଋ ⇔
m
k=n
zk ≤
∞
def n=0
n
⎪
⎩
1
|q| < 1
1−q
ൃࢄ |q| ≥ 1
m
f (z) =
⎝f (z)
͸ࢦ਺ؔ਺
e
z
⎧
⎪
⎪
⎨
z =⎪
n
⎪
⎩
1
։ԁ൘ |z| < 1
1−z
ൃࢄ |z| ≥ 1
f (n) (0)
1
ͷ̩ల։... ܎਺ an =
=
n!
n!
⎞
⎠
z2
zN
z N +1
+ ··· +
+
+ · · · |z| < N ͳΔ N ΛͱΔ
2!
N ! ⎛ (N + 1)!
⎞
N
z2
z
z ⎝
+
+ · · ·⎠ ͜ΕΛԼ‫ͱه‬ൺֱ
1+
N!
N + 1 (N + 2)(N + 1)
⎞
z z 2
|z|N
|z|N ⎝
1
1 + + + · · ·⎠ →
N!
N
N
N ! 1 − z N bk < k=n
thus,
∞
zn
͸೚ҙͷ z ʹର͠ɼ
ʮઈରऩଋʯ
n=0 n!
n+1 zn z
్த͔Β‫ز‬Կ‫਺ڃ‬ͷ‫ͳʹܗ‬Ε͹ͦΕͰ̤̠˰ ߲ൺ zn < ∞ ˱ ෦෼࿨ ͸͋Δෳૉ਺ʹऩଋ͢Δ
m
(*) ‫ج‬ຊྻ zk → 0. ಛʹ m = n + 1 ͱͯ͠ɼ|zn+1 | → 0
k=n+1 |z1 + z2 | ≤ |z1 | + |z2 |: γϡϫϧπɽ̏֯‫ܗ‬ͷลͷ௕͞ʹؔ͢Δੑ࣭
5
∞
n=0
⎛
⎛
|zn | ͕ऩଋ ˰ ऩଋ
ऩଋ൑ఆϧʔϧ͸͍Ζ͍Ζ͋Δ͕ɼඞ༻ʹԠͯ͡ಋೖ
‫ه‬๏
|zk | ≤
zn
n!
1+z+
(*)
⎧
⎪
⎪
⎨
q =⎪
n
n=0
n=0
্‫|( ه‬zn | = rn ) ʹ͔ΒΊͯΑ͘࢖͏ऩଋ൑ఆϧʔϧ
an (z − z0 )n ɽͲͷൣғͰऩଋ͢Δ͔͕ϙΠϯτ
n=0
n+1
∞
∞
ϕΩ‫ ਺ڃ‬f (z) =
k=0
q = 1 ͷ৔߹ʢq = 1 Ͱࣗ໌ʹൃࢄʣ
⎧
n
ϕΩ‫਺ڃ‬
3
n
6
=
|z|
→ 0 ͷ৔߹
n+1
3.1
ʢ‫ʹ਺ڃ‬ର͢Δʣൺ൑ఆ๏
∞
‫਺ڃ‬
zn
∞
n=0
੔ཧɾҰൠԽɿ ൺ൑ఆ๏ ʢൺͷ෼ੳ๏ʹ‫ͮ͘ج‬ʣ
z
n+1 lim
n z n
=p<1˰
n
zn ͸ઈରऩଋ
े෼େͳ N s.t. n > N ˰
z
n+1 z n
k=1
≤q<1 .
zn → 0,
· · · +zN +zN +1 +zN +2 · · ·
· · · +|zN | +|zN +1 | +|zN +2 | · · ·
· · · +|zN | +q|zN | +q 2 |zN | · · ·
ྫɿ
n
zn
:
n!
z
n+1 lim
n z n
z
n+1 z n
=
=p>1˰
n+1 n!z
(n + 1)!z n n
=
zn ͸ൃࢄ
े෼େͳ N s.t. n > N ˰
|z|
→0
n+1
z
n+1 z n
z
n+1 zn → 1 ͷ৔߹
1/(n + 1)
n
=
→ 1ɼ
1/n
n+1
n+1 dt
n k+1 dt
1
≥
=
= ln (n + 1) → ∞
1
k k=1 k
t
t
where
z
n+1 z n
→ 1 Ͱऩଋ͢Δྫɿ
1
2
n=1 n
∞
n2
1/(n + 1)2
→ 1.
=
1/n2
n2 + 2n + 1
n 1
n 1
n k dt
=1+
≤1+
2
2
2
k=1 k
k=2 k
k=2 k−1 t
=1+
≥q>1 .
|zN +k | ≥ q|zN +k−1 | ≥ q 2 |zN +k−2 | ≥ ... ≥ q k |zN |.
zn = 0 (ऩଋ͢Δ‫Ͱ਺ڃ‬͸ͦͷ֤߲͸̌ʹऩଋ)
so, lim
n
lim
n k=1
1
k
n
ʢ‫ʹ਺ڃ‬ର͢Δʣൺ൑ఆ๏
3.2
z
n+1 ଓ͖ɿ
zn = 1 ͷ৔߹͸ʁ ˰ Կͱ΋‫͍ͳ͑ݴ‬ʢ࣍ทʣ
⎡
1
⎤n
1
x−1 ⎦
=1+
=2− ≤2
2
t
−1 1
n
n dt
⎣
n+1
1
1
≤
෦෼࿨͸୯ௐ૿Ճྻ
2
2
k
k=1
k=1 k
n
༗ք୯ௐ૿Ճྻ͸ऩଋ(*)
(*) ্‫ ݶ‬supn an ͸࠷খ্քʢ্քͷ಺ɼ࠷খͳ΋ͷʣͰɼ{an } ͸༗քͳͷͰɼsupn an < ∞.
͞Βʹ ୯ௐ૿Ճྻ {an } ͷ৔߹͸ɼ lim an = supn an ͕੒ཱ
n→∞
্‫ ݶ‬α = supn an ͕࣋ͭੑ࣭ɿ(1) ∀ n an ≤ α, (2) ∀ ∃aN α − < aN
୯ௐੑ͔Β n > N ˰ α − < aN ≤ an ≤ α. ಛʹ |an − α| < .
thus, lim an = α
n→∞
༗ք୯ௐ૿Ճྻͷྫɿ
7
1−
1
2n
: 0,
n≥0
1 3 7 15
, , ,
, ...
2 4 8 16
8
3.3
ϕΩ‫਺ڃ‬ͷऩଋʹؔ͢Δ‫ج‬ຊతࣄ࣮
ϕΩ‫ ਺ڃ‬f (z) =
n
an (z − z0 )n ʹର͠
গͳ͘ͱ΋த৺ z0 Ͱऩଋ
z1 Ͱऩଋɼ|z − z0 | < |z1 − z0 |
˰ z Ͱઈରऩଋ
z1 Ͱൃࢄɼ|z − z0 | > |z1 − z0 | ˰ z Ͱൃࢄ
ऩଋԁɿ ԁपΛআ͖
ऩଋ͢Δ఺Λ಺෦ʹશͯ‫ؚ‬Ή࠷খͷԁ
ʢ಺෦͕ऩଋ͢Δ఺ͷΈ͔ΒͳΔԁͷ͏ͪ࠷େʣ
|z − z0 | = r ऩଋ൒‫ܘ‬
|z − z0 | < r ˰ z Ͱઈରऩଋ
|z − z0 | > r ˰ ൃࢄ
Note: ऩଋԁप্Ͱ͸ɼऩଋɼൃࢄ͸͍Ζ͍Ζ
˰ ԁपΛ‫͍ͳ·ؚ‬։ԁ൘ |z − z0 | < r ͕‫ڵ‬ຯͷର৅ɽ
9
3.4
ϕΩ‫਺ڃ‬ͷઈରऩଋੑ
1. ʮz1 Ͱऩଋ ˰ |z| < |z1 | ͳΔ z Ͱ΋ऩଋʯͷ؆୯ͳઆ໌ʢத৺̌ͷͱ͖ʣ
let
n
an z1n < ∞
|z| < |z1 | ͳΔ z Λ೚ҙʹ‫ݻ‬ఆ
֤߲͸̌ʹऩଋ͢Δ͔Β༗քɽ
|an z1n
| ≤
M
n
z n
z
|an z n | =
|an z1n | ≤ M z1
n
n
n z1
‫ز‬Կ‫਺ڃ‬ͷऩଋੑ͔Βɼઈରऩଋ
2. ऩଋԁ಺෦ʢ։ԁ൘ |z| < rʣͷ೚ҙͷ఺ z Ͱઈରऩଋ
|z| < |z1 | < r ͳΔ ։ԁ൘಺ͷ఺ z1 ΛͱΕΔɽ
z1 Ͱऩଋ͢Δ͔Βɼz Ͱઈରऩଋ
10
3.5
ϕΩ‫ͦͱ਺ڃ‬ͷऩଋɿྫ୊ʢશฏ໘Ͱऩଋʣ
∞
z3 z5
z 2n+1
+ − ··· =
(−1)n
3! 5!
(2n + 1)!
n=0
n+2 2
z
an+2 z
= → 0 (n → ∞) for any z.
߲ൺ (2n + 1)(2n) an z n 3.6
߲ͷ܎਺ൺͷେ͖͞
a
n+1 a n
n+1 an a
(∗) f (z) = z −
೚ҙͷ z Ͱઈରऩଋɽ
ίʔγʔɾΞμϚʔϧͷެࣜʢ‫ʹ਺ڃ‬ର͢Δൺ൑ఆ๏ͷ‫ܥ‬ʣ
L∗ = lim
n ܎਺ൺͷେ͖͞ͷ਺ྻ͕ऩଋ͢Δͱ͢Δ
(*1)
an = 0 (*2)
1. T = |z|L∗ < 1 ͳΔ೚ҙͷ z ʹରͯ͠
n+1 an+1 z
an z n ऩଋ൒‫ܘ‬͸ແ‫ݶ‬େ
lim
n a
n+1 an = |z| → |z|L∗ < 1
‫ʹ਺ڃ‬ର͢Δൺ൑ఆ๏ΑΓ,
ͷ‫ݶۃ‬஋ͷ‫͕਺ٯ‬ऩଋ൒‫ܘ‬
n
|an zn | ͸ऩଋɽ
n
an zn ͸ઈରऩଋ
ಛʹ L∗ = 0 Ͱɼ೚ҙͷ z ʹର͠ |z|L∗ = 0 < 1ɽશฏ໘Ͱऩଋ
ίʔγʔɾΞμϚʔϧ
∗
2. T = |z|L > 1ɿे෼େͳ n > N ʹର͠ɼ߲ൺ
(*) ̏֯ؔ਺ͷ̩ల։ɽ‫ ਺ؔح‬f (−z) = −f (z) Ͱɼsin z
n+1 an+1 z
an z n ൃࢄʢ֤߲ an zn ͷେ͖͞͸‫ڱ‬ٛͷ୯ௐ૿Ճྻʣ
3. ͕ͨͬͯ͠ɼ‫ݶۃ‬஋ L∗ ʹର͠ɼऩଋ൒‫ ܘ‬r = 1/L∗
(*1)
L∗
a
n+1 n+1 z
͕ ∞ ʹൃࢄͷ৔߹ɿ߲ൺ ͸‫ݶ‬Γͳ͘େʹɽ
an z n |an+1 z n+1 | > K|an z n | whenever n > N for some N
֤߲͸̌ʹऩଋͤͣɼ೚ҙͷ z = 0 ʹର͠ൃࢄɽऩଋ൒‫̌ʹܘ‬
(*2) ൺΛͱΓʹ͍͘৔߹͸ɼࠜ൑ఆ๏ͳͲ
11
12
>1
3.7
ʮࠜ൑ఆ๏ʯ
ɿ܎਺ͷઈର஋ͷ n ৐ࠜʣ
3.8
n
(1) े෼େͳ N Ҏ্Ͱɼ |zn | ≤ q < 1 ʢn > N ʣ
n
˰
z ͸ઈରऩଋ
n
|zn | ≤ q n ɽऩଋ౳ൺ‫Ͱ਺ڃ‬཈͑ΒΕΔ͔Β ઈରऩଋ
n
L = lim
|zn | < 1 ˰
n
n
zn ͸ઈରऩଋ
|zn | ͸ɼN Λे෼େʹͱΕ͹ɼ
̍ΑΓਅʹখ͞ͳ஋ q0 ͷۙ๣ (q0 − , q0 + ) ʹूத͢Δɽ
q = q0 + ͱͯ͠ɼ(1) Λ࢖͑Δ
lim n |an (z − z0 )n | = |z − z0 | = n→∞
lim n |an | < 1
n→∞
|z − z0 | < 1/ ͳΒઈରऩଋ
√
(2) ্‫ه‬ͷ߹Θٕͤɿ ෦෼ྻ ʹର͢Δ n an z n ͷ‫ݶۃ‬஋Λ࢖͏ํ๏
n
f ɿ z0 Λ‫ؚ‬ΉྖҬ D Ͱղੳత
ӈਤͷ։ԁ൘಺ͷ z Λ‫ؚ‬Ήด࿏ C ɽ
1 f (w)
dw
ੵ෼ެࣜΑΓ f (z) = g(z) =
def 2πi C w − z
g(z) ʹର͢Δࣜม‫Ͱܗ‬ɺ
f (k) (z0 )
(z − z0 )k + Rn (z),
k!
k=1
(z − z0 )n+1 f (w)
Rn (z) =
dw
C (w − z0 )n+1 (w − z)
2πi
→ 0 (as n → ∞)
n f (k) (z )
0
(z − z0 )k ... (*)
g(z) = n→∞
lim
k!
k=0
g(z) =
n
ςΠϥʔͷެࣜɾ‫਺ڃ‬
1 + (−1)n +
1
zn
2n
|z|
2
n
n
‫ʹྻ਺ۮ‬ର͠ɼ̽৐ࠜΛͱΔͱɼ (2 + 1/2 )|z|n → |z|
1
ln (2 + 1/2n ) → 0
n
े෼େͳ N Ҏ্Ͱɼ n |an z n | ≤ |z| ͔ͩΒɼ
|z| < 1 Ͱ͋Ε͹ an z n ͸ઈରऩଋ
‫ʹྻ਺ح‬ର͠ɼ̽৐ࠜΛͱΔͱɼn |z|n /2n =
n
(*) ͸ |z − z0 | < r ͳΔ೚ҙͷ z ʹର͠ʢC ʹґଘͤͣʹʣ੒ཱ͠ɼ
͔ͭɼͦͷൣғͰ f (z) = g(z)
f (z) =
f (k) (z0 )
(z − z0 )k ... f ͷϕΩ‫਺ڃ‬ද‫ݱ‬
k!
k=0
౳߸ͷ੒ཱ͢Δൣғɿ |z − z0 | < r
∞
ۙࣅ͸௨ৗখ͞ͳԁ൫Ͱߟ͑ΔͷͰɼෳૉղੳؔ਺ͷ৔߹͸ৗʹ͜
͏ͨ͠ද‫͕ݱ‬Մೳͩͱߟ͑ͯྑ͍
n
܎਺ {an }n ʹର͠ɼn |an | ͕ऩଋ͢Δ෦෼ྻʹ෼ղͰ͖ɼ͔ͭͦ
ΕΒ࠷େͷ‫ݶۃ‬஋Λ ͱ͢ΔͱɼϕΩ‫਺ڃ‬
n
n
൒‫ܘ‬͸ 1/ Ͱ͋Δ
13
a (z − z0 )n ͷऩଋ
14
3.9
࣍ʹɼRn (z) → 0 as n → ∞ ͷ‫ূݕ‬ɿ
̩ެࣜͷূ໌ུ֓
f (w)
dw ... z Λม਺Խ͠ɺ(z − z0 ) ͷද‫ݱ‬Λ࡞Δ
−z
f (w)
f (w)
dw =
=
z − z0 dw
C (w − z0 ) − (z − z0 )
C
(w − z0 ) 1 −
w − z0
1 − q2 + q2
q2
1
2
=
=1+q+
(z − z0 ) ͸෼ࢠʹ‫ݱ‬ΕΔɻͦ͏ͳΔ΂ࣜ͘ม‫ܗ‬ɿ
1−q
1−q
1−q
⎛
⎞
z − z0 2
f (w) ⎜
z
−
z
0
w − z0 ⎟
⎜
⎟
+
2πig(z) =
⎜1 +
⎟ dw
w − z0 1 − z − z0 ⎠
C (w − z0 ) ⎝
w − z0
z − z0
(z − z0 )2
f (w)
1+
+
dw
=
2
w − z0 (w − z0 ) − (z − z0 )(w − z0 )
C (w − z0 )
2
z − z0
(z − z0 )
f (w)
1+
+
dw
=
w − z0 (w − z0 )(w − z)
C (w − z0 )
f (w)
f (w)
f (w)
dw
dw + (z − z0 )
dw + (z − z0 )2
=
C w − z0
C (w − z0 )2
C (w − z0 )2 (w − z)
2πi g(z) =
Cw
ߴ֊ͷੵ෼ެࣜ
C (w
f (w)
Λ̢̡ͰධՁ
dw
− z0 )n+1
(w − z)
f
(w)
<M
|w − z0 | = rɺ
w − z ʢw = z ɺ༗քͳྖҬͷ࿈ଓؔ਺ʹΑΔ૾͸༗քʣ
೚ҙʹ‫ݻ‬ఆͨ͠ԁ಺ͷ z Λߟ͑Δ
(z − z0 )n+1 f (w)
Rn (z) =
dw
C (w − z0 )n+1 (w − z)
2πi
|z − z0 |n+1 f (w)
|Rn (z)| =
dw
n+1
C (w − z0 )
2π
(w
−
z)
n+1 |z − z0 |
f
(w)
≤
dw
(L = 2πr)
2πrn+1 C w − z z − z n+1
|z − z0 |n+1
0
2πrM = M r → 0 as |z − z0 | < r
<
2πrn+1
r f (n) (z0 )
f (w)
1 =
dw ΑΓɼ
n!
2πi C (w − z0 )n+1
g(z) = f (z0 ) + f (z0 )(z − z0 ) +
f (w)
(z − z0 )2 dw
2πi
C (w − z0 )2 (w − z)
n ≥ 2 ͷ৔߹͸ɺ(z − z0 )n+1 Λ෼ࢠʹ͢ΔͨΊͷࣜม‫ͯ͠ͱܗ‬Լ‫ه‬Λ࢖͏ɿ
1
1 − q n+1 + q n+1
q n+1
=
= 1 + q + · · · qn +
1−q
1−q
1−q
15
16
‫Ͱ఺ݪ‬ͷల։ɿϚΫϩʔϦϯల։ͷྫ
1
f (z) =
= (1 − z)−1
1−z
f (z) = (1 − z)−2 , f (z) = 2(1 − z)−3 ,
3.10
f (z) = 2 ∗ 3(1 − z)−4 , ...
f (n) (0)
=1
f (n) (z) = n!/(1 − z)n+1 ,
n!
ϕΩ‫਺ڃ‬ͷऩଋҬͰɼ‫ ͱ਺ڃ‬f ͸Ұக
1
= z n (|z| < 1)
1−z
n
౳߸͸த৺͔Βಛҟ఺Λ‫࠷͍ͳ·ؚ‬େͷ։ԁ൘
17
3.11
ྫʢର਺ɼओ஋ʣ
f (z) = Ln (1 + z) Λ z = 0 Ͱల։͢Δ
(Note: Ln z ͸ z = 0 Ͱະఆٛ‫ނ‬ɺϚΫϩʔϦϯ ల։Ͱ͖ͳ͍)
f (0) = 0,
f (z) = 1/(1 + z), f (0) = 1
f (2) (z) = −(1 + z)−2 , f (2) (0) = −1, a2 = −1/2
f (3) (z) = 2(1 + z)−3 , f (3) (0) = 2, a3 = 2/(2 ∗ 3) = 1/3
f (4) (z) = −3 ∗ 2(1 + z)−4 , f (4) (0) = −3 ∗ 2
...
z2 z3 z4
Ln (1 + z) = z − + − + · · ·
2
3
4
(|z| < 1)
18
3.12
ର਺ؔ਺ͷϕΩ‫਺ڃ‬ද‫ݱ‬ɼϕΩ‫਺ڃ‬ͷऩଋ൒‫ܘ‬
ओ஋ͷର਺ f (z) = Ln z, z0 = i − 1
√
√
3
f (i − 1) = Ln 2e3/4πi = ln 2 + πi
4
f (z) = z −1 , f (z) = −z −2 , f (3) (z) = 2!z −3 , f (4) (z) = −3!z −4 , · · ·
f (n) (z0 )
(−1)n−1
f (n) (z) = (−1)n−1 (n − 1)!z −n . ܎਺ an =
=
(z0 )−n
n!
n
√
z − i + 1 1 (z − i + 1)2
3
−
f (z) = ln 2 + πi +
+ · · · (|z − i + 1| < 1)
4
i−1
2 (i − 1)2
f ͷ T ϕΩ‫਺ڃ‬දࣔ ͸ |z − i + 1| < r1 = 1 ͕ͩɼϕΩ‫਺ڃ‬ͷऩଋ൒‫ܘ‬͸
√
|an |
n + 1 |z0 |n+1
1
=
=
1
+
|z
|
→
|z
|
=
2
0
0
|an+1 |
n
|z0 |n
n
ln z ͱͯ͠ɼภ֯Λ 0 < arg z ≤ 2π ʹͱΔର਺ؔ਺ f2 (z)
√
3
̌࣍ͷ߲ f (i − 1) = f2 (i − 1) = ln 2 + πi
4
f (z) = f2 (z) = 1/z ΑΓɼf (z) ͱ
f2 (z) ͷ̩‫਺ڃ‬ͷ̽࣍
(−1)n−1
(z0 )−n ͸ಉ͡
ͷ߲ ܎਺ an =
n
3.13
M ల։ͷྫɿ ߲̎‫਺ڃ‬
f (z) = (1 + z)−m
f (1) (z) = (−m)(1 + z)−m−1 , f (2) (z) = (−m)(−m − 1)(1 + z)−m−2 , ...
f (n) (z) = (−m)(−m − 1) · · · (−m − n + 1)(1 +⎛z)−m−n⎞
f (n) (0) (−m)(−m − 1) · · · (−m − n + 1) ⎜ −m ⎟
⎠
=
=⎝
n!
n!
n
√
2
−m
n
:
ಛʹɺ
⎛
(1 + z)−m =
⎜
⎝
n
−m
0
−m
n
m
n
=
m!
m(m − 1)...(m − n + 1)
ʹशͬͯఆٛ
=
n!(m − n)!
n!
= 1 ͱ໿ଋ.
⎞
⎟ n
⎠z
(|z| < 1)
݁‫ہ‬ɼ (1 + z)−m = 1 − mz +
f2 (z) = f (z) ͷ̩‫ͱ਺ڃ‬ಉҰͷ‫਺ڃ‬
√
౳߸͸ |z − i + 1| < 2 ͰϕΩ‫਺ڃ‬ͷऩଋԁ൫ͱҰக
19
20
m(m + 1) 2 m(m + 1)(m + 2) 3
z −
z + ···
2
3!
3.14
ྫ ߲̎‫਺ڃ‬ʢcontinued 1ʣ
1
ͷϚΫϩʔϦϯ‫਺ڃ‬
(z − 2)5
ߟ͑ํɿ ‫ܗ‬͸ (1 + z)−m =
z−2=2
(z − 2)
−5
⎛
⎜
⎝
n
−m
n
So, (z − 2)
=
⎛
∞
⎜
⎝
n=0
−5
n
(1 + z)−m =
⎞
⎟ n
⎠z
(|z| < 1)
⎞
⎟
n+1
⎠ (−1)
zn
2n+5
⎛
⎜
⎝
−5
n
⎞
⎟
⎠
⎛
⎜
⎝
n
−m
n
⎞
⎟ n
⎠z
(|z| < 1)
f (z) = (2 + z)−2 + 2(z − 3)−1
z
z
− 1 = −2 1 −
2
2
−5
z
1
= (−2)−5 1 + −
=− 5
2
2 n
−5
ྫ ߲̎‫਺ڃ‬ʢcontinued 2ʣ
3.15
−
z
2
n
(| − z/2| < 1 ⇔ |z| < 2)
z0 = 1 Λத৺ʹͨ͠ల։ Λߦ͏ͱ͠ɺ(z − 1) ͷࣜʹ͢Δɿ...(∗2)
z−1
z−1
2 + z = 3 + (z − 1) = 3 1 +
, z − 3 = (z − 1) − 2 = −2 1 −
2
3
z − 1 −2
z − 1 −1
f (z) = 3 1 +
−2 2 1−
3
2
⎞
⎛
n
z−1 n
1 ⎜ −2 ⎟ z − 1
⎠
⎝
=
−
9n
3
2
n
n
⎡
⎤
(−1)n (n + 1)
=
− 2−n ⎦ (z − 1)n
3n+2
n
z − 1
z − 1
< 1 ͔ͭ < 1 ΑΓ |z − 1| < 2
ऩଋൣғɿ 2 3 ⎣
(*2) ͍Ζ͍Ζͳม‫͕ܗ‬Մೳ͕ͩɺ͜͜Ͱ͸؆୯ͳ (1 − z)−1 =
(*3)
21
n
(−2)(−2 − 1)...(−2 − n + 1)
n(n − 1)...3.2.1
n 2(2 + 1)...(2 + n − 1)
= (−1)n (n + 1)
= (−1)
n(n − 1)...3.2.1
−2
n
=
22
zn
ϕΩ‫਺ڃ‬ͷ߲ผඍ෼
3.16
f (z) ⇔
def n≥0
3.17
an z n ͷ֤߲ an z n Λඍ෼ͯ͠࡞ͬͨϕΩ‫਺ڃ‬
g(z) ⇔
def n≥1
nan z
n−1
=
ϕΩ‫਺ڃ‬͸ऩଋԁ಺෦Ͱղੳత
ϕΩ‫਺ڃ‬͸ऩଋԁ಺෦Ͱඍ෼ՄೳͰͦͷಋؔ਺͸༠ಋ‫ͱ਺ڃ‬Ұக
ূ໌͸૬౰ʹٕ޼తɽΑͬͯεΩοϓ
(m + 1)am+1 z
m
f (z) ⇔ a0 + a1 z + a2 z 2 + · · · + an z n + · · ·
m≥0
def
ΛϕΩ‫ ਺ڃ‬f (z) ͷʮ༠ಋ‫਺ڃ‬ʯͱ͍͏ɽ̎ͭͷ‫਺ڃ‬ͷऩଋ൒‫ܘ‬͸ಉ͡
a
1
n+1 =
lim
͕ଘࡏ͢Δ৔߹͸
n
an r
m + 2 am+2 (m + 2)am+2 =
(m + 1)am+1 m + 1 am+1 →
=
1
r
as
m+2
→1
m+1
1
= 1 + z + z 2 + z 3 + ... |z| < 1
1−z
⎜
⎝
−2
0
⎞
⎛
⎟
⎠
⎜
⎝
+
−2
1
⎞
⎛
⎟
⎠ (−z)
⎜
⎝
+
−2
2
⎞
⎛
⎟
2
⎠ (−z)
⎜
⎝
n = 2 : I2 = 2zΔz + (Δz)2 /Δz − 2z
n = 3 I3 = (3z 2 Δz + 3z(Δz)2 + (Δz)3 )/Δz − 3z 2
n = 4 I4 = (4z 3 Δz + 6z 2 (Δz)2 + 4z(Δz)3 + (Δz)4 )/Δz − 4z 3
...
→ 0 as Δz → 0 Λࣔ͢
= Δz
= Δz (3z + Δz)
= Δz 6z 2 + 4zΔz + (Δz)2
k
z ͷ܎਺͕ෳ਺ͷ n ʹ‫ͯͬލ‬ग़‫)*( ݱ‬
+
−2
3
⎟
3
⎠ (−z)
n≥0
⎡
⎤
⎢
⎢
In = Δz ⎢
⎣
⎥
⎥
⎥
⎦
⎞
߲ผඍ෼ͰಘΒΕΔ‫਺ڃ‬ɿ ༠ಋ‫਺ڃ‬ɽ‫ݩ‬ͷ‫਺ڃ‬ͷඍ෼
23
(z + Δz)n − z n
− nz n−1
Δz
(z + Δz)n−2 + 2z(z + Δz)n−3 + ... + (n − 1)z n−2
࣍਺͕ n − 2 ͱͳΔ૊Έ߹ΘͤͰɼn ‫ݸ‬ͷࣜ
܎਺ͷ࠷େ஋͸ n − 2
|z|, |z +
Δz| ≤ r⎛0 < r ͳΔ r ΑΓখ͞ͳ r0 ͕ͱΕΔ
⎞
+ ...
(−2)
(−2)(−3)(−4)
(−2)(−3)
=1+
(−z) +
(−z)2 +
(−z)3 + ...
1
2
3∗2
n
= 1 + 2z + 3z 2 + 4z 3 + ...
z ͷ֤߲Λඍ෼
1+n
ӈลͷऩଋ൒‫ ܘ‬lim
= 1 ΑΓ̍
n
n
্‫ )*( ه‬ͷ໰୊Λճආ͢ΔͨΊʹɼ͔֬ʹʮٕ޼తʯͳԼ‫ه‬ͷࣜม‫ܗ‬Λߦ͏ɿ
−1
= (1 − z)−2 . ͦͷ M ల։͸Լ‫ه‬
f (z) = −
(1 − z)2
⎛
def
f (z + Δz) − f (z)
an I n =
− g(z) =
Δz
n≥2
a
m+2 lim
m a
m+1
Ұൠͷ৔߹ͷূ໌͸ ϖʔδ 3.18 ͷ‫٭‬஫
ྫɿ f (z) =
g(z) ⇔ a1 + 2a2 z + · · · + an nz n−1 + · · · f ͷ༠ಋ‫਺ڃ‬ɽf ͱಉҰͷऩଋ൒‫ ܘ‬r
Αͬͯɼ n≥2
n≥2
an In ≤ |Δz| ⎝RHS =
|an |n(n − 1)r0n−2 ⎠
RHS ͸ f, g ͱಉҰͷऩଋ൒‫ ܘ‬r Λ࣋ͭ g ͷ༠ಋؔ਺
h(z) ⇔
def
n(n − 1)an z n−2
f ͷ̎ճඍ෼
n≥3
্͕࣮࣠ͷ z = r0 Ͱઈରऩଋ͢Δ͜ͱΛ͔ࣔࣜͩ͢Βɼऩଋ͢Δɽ
f (z + Δz) − f (z)
ͭ·Γɼ
Δz
− g(z)
= n≥2
24
an In ≤ |Δz||h(r0 )| → 0 as Δz → 0
3.18
ϕΩ‫਺ڃ‬ͷ߲ผඍ෼ɿ·ͱΊ
3.19
ϕΩ‫Ͱ਺ڃ‬ఆٛ͞Εͨؔ਺ f Λߟ͑Δ
f (z) ⇔
def n≥0
an z n
f (z) = a0 + a1 z + a2 z 2 + · · · (|z| < r) ͱ͢Δ
z
a1
a2
߲ผੵ෼ an wn dw Ͱ F (z) = a0 z + z 2 + z 3 + · · · Λఆٛ
0
2
3
f ͷӈล ͸ F ͷ༠ಋؔ਺͔ͩΒɼf ͷӈลͱ F ͷऩଋ൒‫ܘ‬͸ಉ͡Ͱɼ
ಛʹɼF (z) = f (z) for |z| < rɽ
ల։த৺ ̌
ऩଋ൒‫ ܘ‬r
͔̌Β‫͢ࢄൃͯݟ‬Δ఺Λ಺෦ʹ‫࠷͍ͳ·ؚ‬େͷԁ
ൃࢄ͢Δ఺Ͱ͸ f (z) ͸ະఆٛ
|z| < r ಺ͷ z ʹର͠
̍ɽऩଋԁ಺Ͱ f (z) ͸ղੳతɽ̩ఆཧͱ܎਺ͷҰҙੑ͔Βɼ
f (z) =
f (n) (0) n
z (|z| < r)
n!
n≥0
ie an =
f (n) (0)
n!
̎ɽ߲ผඍ෼ɿ֤߲Λඍ෼ͯ͠ಘΒΕΔ‫਺ڃ‬
g(z) ⇔
def
ϕΩ‫਺ڃ‬ͷ߲ผੵ෼
f (n) (0) n−1
ʹ
z
(m = n − 1)
n≥1 (n − 1)!
z
0
f (w)dw
z
0 n
=
F (z) − F (0) = F (z)
=
‫਺ؔ࢝ݪ‬
F ͷఆٛ
an wn dw =
z
n
0
an wn dw
(|z| < r)
f (m+1) (0) m
z (|z| < r)
m!
m≥0
f (z) ͷ ̢‫਺ڃ‬
an+1 ͕ൃࢄ͢Δ৔߹΋‫ؚ‬Ίɼ΋ͱͷϕΩ‫਺ڃ‬
a ίʔγʔɾΞμϚʔϧͷެ͕ࣜ࢖͑ͳ͍৔߹ɼͭ·Γɼ
ͱ༠ಋ‫਺ڃ‬ͷऩଋ൒‫͕ܘ‬ಉ͡Ͱ͋Δ͜ͱͷূ໌
n
an z n ͕ |z| < r Ͱऩଋ ˰
n≥0
|z| < r1 < r ͳΔ r1 Λద౰ʹͱΔɽ
|z1 | = r1 ͳΔ z1 Ͱऩଋ͢Δ͔Β |an z1n | = |an |r1n < M ʢ֤߲͸༗քʣ
M
M |z n |
so,
n|an z n−1 | ≤
n n |z n−1 | ≤
n
r1
|z| n≥1 r1n
n≥1
n≥1
ӈลͷ‫ʹ਺ڃ‬ର͠
Αͬͯɼ
| ୈ (n+1) ߲ |
(n + 1)|z|n+1 r1n
=
=
|ୈ n ߲|
n|z|n
r1n+1
n+1
n
|z|
|z|
→
<1
r1
r1
nan z n−1 ͸ઈରऩଋɽͦͷऩଋ൒‫ܘ‬Λ ΞμϚʔϧͷެ͔ࣜΒ‫ٻ‬ΊΔ
n≥1
nan z n−1 ʹର͠
n ൪໨ͷ܎਺
n|an |
=
ɽͦͷ‫ݶۃ‬ɿ
n + 1 ൪໨ͷ܎਺
(n + 1)|an+1 |
|an |
lim
n→∞ |an+1 |
an z n ͷऩଋ൒‫ܘ‬ʂ
n
25
26
z
2
z1
∞ z
n=0 0
an wn dw
f (w)dw ͳͲ΋ಉ༷
3.20
߲ผੵ෼ͱ࣮਺ੵ෼ͷྫ
ྫɿ ԋ໰ 3.4.11 S(z) =
z
0
3.21
sin t2 dt (∗)
f (z) = ez , f (n) (z) = ez , e0 = 1ɽ ez =
sin z 2 ͸શฏ໘ͰղੳతͰϕΩ‫ʹ਺ڃ‬ల։Մ
z3 z5
sin z ͷ̩ల։ sin z = z − + − · · · (શฏ໘)
3! 5!
⎛
⎞
2 3
z
z
(w
z3
(w2 )5
z7
z 11
)
2
2
⎝
sin w dw =
(w ) −
+
− · · ·⎠ dw =
−
+
···
0
0
3!
5!
3
7 3! 11 5!
ʢશฏ໘ʣ
ಛघͳ৔߹ͱͯ͠ z ͕࣮਺ x ͷ৔߹͸࣮਺ੵ෼ͷϕΩ‫ࣅۙ਺ڃ‬
(*) ࣮਺ੵ෼
x
0
f (t)dt Λෳૉੵ෼Ͱ‫ٻ‬ΊΔɿ
্࣮࣠ͷ۠ؒ ˰ ෳૉฏ໘্ͷઢ෼ɽಛʹੵ෼࿏
z(t) = t, dz = dt, 0 ≤ t ≤ x,
[0,x]
f (z)dz =
x
0
f (t)dt
‫఺ݪ‬ɼx = x + 0i Λ‫ؚ‬Ή୯࿈݁ྖҬͰղੳత
f (z) ͷ‫ ਺ؔ࢝ݪ‬F (z) ʹର͠ɼ
x
0
ྫ̎ʢࢦ਺ؔ਺ɼ̏֯ؔ਺ʣ
f (t)dt = F (x) − F (0)
cos z, sin z
eiz + e−iz
eiz − e−iz
, sin z =
2
2i
(iz)n
(−iz)n
iz
−iz
+
2 cos z = e + e =
n!
n!
n
n
2n
2n+1
2n
(iz)
(iz)
(−iz)
(−iz)2n+1
=
+
+
+
(2n)!
n (2n)!
n (2n + 1)!
n
n (2n + 1)!
2n
z
(શฏ໘)
= 2 (−1)n
(2n)!
n
cos z =
z 2n
(2n)!
n
z2 z4
= 1 − + − ···
2! 4!
ʢશฏ໘ʣ
cos z =
(−1)n
߲ผੵ෼ or
(cos z) = − sin z
−→
←−
߲ผඍ෼
ࢦ਺΍ࡾ֯ؔ਺ͳͲɼෳૉ਺ʹ֦ுͨ͠ղੳతͳؔ਺ʹର͠ɼ
‫਺ڃ‬ల։ʹΑΔ࣮਺ੵ෼ͷۙࣅ‫ ࢉܭ‬Λద༻Մ
27
zn
f (n) (0) n
ʢશฏ໘ʣ
z =
n!
n≥0
n≥0 n!
28
z 2n+1
(2n + 1)!
n
z3 z5
= z − + − ···
3! 5!
ʢશฏ໘ʣ
sin z =
(−1)n
ྫ̏ʢ୅ೖͱඍ෼ʣ
1
1
=
୅ೖ:
= (−z 2 )n = (−1)n z 2n ,
1 + z2
1 − (−z 2 )
n
n
| − z 2 | < 1 ˱ |z| < 1
3.22
ಋؔ਺͕‫਺ڃ‬ల։ࡁ ɻ߲ผੵ෼Ͱ΋ͱͷؔ਺ͷ‫਺ڃ‬Λ‫ٻ‬ΊΔ
1
= (−1)n z 2n (|z 2 | < 1 ⇔ |z| < 1)
2
1+z
n
z
z 2n+1
n z 2n
f (w)dw = (−1)
w dw = (−1)n
(|z| < 1)
0
0
2n + 1
n
n
f (z) = tan−1 z (∗), f (z) =
(*) ‫ڭ‬Պॻ̐̒ϖʔδ ໰ 1.8.30 (e) ʢ‫ٯ‬ਖ਼઀ͷؔ਺‫ܗ‬ʣ
w = tan−1 z ˱ z = tan(w = a + bi) =
sin w
eiw − e−iw
ie−2iw − i
=
=
cos w
i(eiw + e−iw )
1 + e−2iw
ie−2iw − i
−2iw
i+z
1 + e−2iw = 2ie
=
= e−2iw
−2iw
ie
−i
i−z
2i
i−
1 + e−2iw
i+z
i i+z
1
ln
= ln
... ln ͷଟՁੑΑΓෳ਺ͷ஋
Hence, w =
−2i i − z
2 i−z
ln ͷภ֯ͷબ୒๏ʹґଘͤͣʹ (ln z) = 1/z
i+
i i−z i+z i i+z g f g − f g
ln
=
=
2 i−z
2 i+z i−z
f
f2
z−i
1
1
i i − z (i − z) + (i + z)
=
=
=
2 i+z
(i − z)2
(i + z)(z − i)2
(z + i)(z − i) z 2 + 1
1
ie (tan−1 z) = 2
z +1
Note: ඍ෼Λͱͬͯ n ͷҧ͍͕ʮফ͑Δʯ͕ɼੵ෼͢Δͱओ஋ͱͦΕҎ֎ͷҧ͍͕෮‫͢׆‬Δɽؔ࿈໰
୊͕ɼ໰ 3.4.14 ʢ119 pageʣʹɽ
29